Portfolio optmization

In portflio optimization, the relevant measures are the rewards of the strategy and the risks are the standard deviation of returns.

• Mean variance analysis was suggested by Harry Markoqitz , who won the Nobel prize in economics in 1990.
• The first method of portfolio optimization asks for a threshold rmin, and outputs a portfolio with expected return more than rmin and minimizes variance as far as possible.
• The second method computes the mean and standard deviation of each security. Note the fact that the portfolio is a convex combination of the given securities. The mean return of the porfolio is the convex combination of the means. The variance of the portfolio is a combination of elements in the variance covariance matrix of the securities. Hence computing these statistics is enough to find the variance minimizing portfolio with expected return above the threshold.
• Now we want to penalize only deviation below the mean return. Hence we define di = max(0, mean - return in this scenario) and try to minimize the sum of squares of di.
  
• Another thing to note is that when we are only concerned with downside risk we need not sqare it. So we minimize average downside risk. Note that in all of these we come with one eficient portflio. The reults are similar in most cases. For nrmal distributions they are identical.